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arXiv · 2103.06163

On Certain Probabilistic Properties of Polynomials over the Ring of $p$-adic Integers

Antonio Lei·Antoine Poulin
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Abstract

In this article, we study several probabilistic properties of polynomials defined over the ring of $p$-adic integers under the Haar measure. First, we calculate the probability that a monic polynomial is separable, generalizing a result of Polak. Second, we introduce the notion of two polynomials being strongly coprime and calculate the probability of two monic polynomials {being} strongly coprime. Finally, we explain how our method can be used to extrapolate other probabilistic properties of polynomials over the ring of $p$-adic integers from polynomials defined over the integers modulo powers of $p$.

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Antonio Lei, Antoine Poulin. 2021-03-10. On Certain Probabilistic Properties of Polynomials over the Ring of $p$-adic Integers. https://doi.org/10.1080/00029890.2020.1736468

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